Externalities, Nonconvexities, and Fixed Points

In this example, let the ith player’s strategy set be given by P i with typicalelement μ i (i.e., X i := P i ). We will refer to the probability measure μ i ∈ P i as arandom i-sender network (i.e., each player’s strategy is to choose a random sendernetwork which in turn determines the realized i-sender network g i ∈ G + iaccording tothe probability measure μ i ). We note that μ i is definedontheBorelσ-field, B(G + i ),in G + igenerated by the h K -open sets.Next letP := P 1 ×···×P mbe the compact metric space of product probability measures defined on the productσ-field, B(G 1 ) ×···×B(G m ), with typical elementμ = μ 1 ×···×μ m .We note that μ ∈ P has support contained in G. We will refer to μ ∈ P as a randomnetwork.Taking P as the parameter space, we can think of the parameter μ e := (μ e i , μe −i ) ∈P, as players’ consensus probability beliefs concerning the likelihood with which variousnetworks might emergence in equilibrium (i.e., Z := P). We will assume that eachplayer’s feasible set of random networks is given by a w ∗ -continuous correspondence,Φ i (·) :P −i → P w ∗ fc(P i ),from consensus probability beliefs into the collection of all nonempty, w ∗ -closed, andconvex subsets of random i-sender networks, denoted by P w ∗ fc(P i ).Thus, Z = X = P, satisfying [A-1](2) and (3) - and thus by Theorem 4, Z andX are dendrites.2.7 Descriptive Summary of Main ResultsLet {G z : z ∈ Z} be any parameterized collection of strategic form games satisfyingassumptions [A-1]. Under [A-1] for each parameter value (or externality value) z ∈ Z,the corresponding strategic form game, G z , has a nonempty, compact subset of Nashequilibria, N (z). Our main objective is to show that the Nash correspondence, z →N (z), is approximable. Because the Nash correspondence may have nonconvex - andpossibly disconnected - values, the problem is difficult and requires a new approach(again see De Blasi and Myjak, 1986, for an extensive summary of results on thegraphical approximation of convex-valued correspondences by continuous functions).Here is a descriptive summary of our results, assuming [A-1] throughout. In ourdescription we will use the terms USCO and CUSCO. An USCO is an upper semicontinuous,set-valued mapping with nonempty, compact values, while a CUSCO is anupper semicontinuous, set-valued mapping with nonempty, compact and connectedvalues (e.g., see Hola and Holy, 2009).(1) (Properties of Nash Correspondences) We show that the Nash USCO, N (·),is a composition of two USCOs: an USCO mapping Ky Fan sets into the collectionof sets of Nash equilibria (the KFC mapping) composed with an USCO mapping Z15